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Properties

Label 2-504-56.27-c1-0-33
Degree $2$
Conductor $504$
Sign $-0.835 - 0.549i$
Analytic cond. $4.02446$
Root an. cond. $2.00610$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.733 − 1.20i)2-s + (−0.924 + 1.77i)4-s − 1.12·5-s + (2.11 + 1.59i)7-s + (2.82 − 0.181i)8-s + (0.826 + 1.36i)10-s − 5.11·11-s − 5.88·13-s + (0.375 − 3.72i)14-s + (−2.28 − 3.28i)16-s − 3.31i·17-s − 7.49i·19-s + (1.04 − 2.00i)20-s + (3.74 + 6.18i)22-s + 1.73i·23-s + ⋯
L(s)  = 1  + (−0.518 − 0.855i)2-s + (−0.462 + 0.886i)4-s − 0.504·5-s + (0.798 + 0.601i)7-s + (0.997 − 0.0641i)8-s + (0.261 + 0.431i)10-s − 1.54·11-s − 1.63·13-s + (0.100 − 0.994i)14-s + (−0.572 − 0.820i)16-s − 0.804i·17-s − 1.71i·19-s + (0.233 − 0.447i)20-s + (0.798 + 1.31i)22-s + 0.362i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.835 - 0.549i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.835 - 0.549i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $-0.835 - 0.549i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ -0.835 - 0.549i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0185787 + 0.0621088i\)
\(L(\frac12)\) \(\approx\) \(0.0185787 + 0.0621088i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.733 + 1.20i)T \)
3 \( 1 \)
7 \( 1 + (-2.11 - 1.59i)T \)
good5 \( 1 + 1.12T + 5T^{2} \)
11 \( 1 + 5.11T + 11T^{2} \)
13 \( 1 + 5.88T + 13T^{2} \)
17 \( 1 + 3.31iT - 17T^{2} \)
19 \( 1 + 7.49iT - 19T^{2} \)
23 \( 1 - 1.73iT - 23T^{2} \)
29 \( 1 - 5.88iT - 29T^{2} \)
31 \( 1 + 6.04T + 31T^{2} \)
37 \( 1 - 1.65iT - 37T^{2} \)
41 \( 1 + 1.45iT - 41T^{2} \)
43 \( 1 - 1.79T + 43T^{2} \)
47 \( 1 + 5.56T + 47T^{2} \)
53 \( 1 - 3.62iT - 53T^{2} \)
59 \( 1 + 0.767iT - 59T^{2} \)
61 \( 1 + 0.317T + 61T^{2} \)
67 \( 1 + 6.56T + 67T^{2} \)
71 \( 1 + 10.1iT - 71T^{2} \)
73 \( 1 - 6.63iT - 73T^{2} \)
79 \( 1 + 3.01iT - 79T^{2} \)
83 \( 1 + 16.0iT - 83T^{2} \)
89 \( 1 - 8.08iT - 89T^{2} \)
97 \( 1 - 0.357iT - 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.52138317139491163228544094034, −9.506643478050139793822406973441, −8.772121293668798921058407605287, −7.62926348371490908629333882575, −7.36339418114316939597170236814, −5.15720761060947025636201993848, −4.74704079499879852478674773485, −3.01387963644034268096922165686, −2.20100219585388646098340384008, −0.04288779475390481150725602100, 2.01537748307467948451820257121, 4.03989871771842700026233712829, 5.01324867783520444867986795372, 5.87347062914771668653808248135, 7.32338167311367757717820598476, 7.81236175111343419082344739497, 8.324459018197478035551441534504, 9.844933333394492765741592958235, 10.27808809270374876273905309307, 11.17755950721785251500700298932

Graph of the $Z$-function along the critical line